Preheat the oven to 350°F (175°C). Spray a 13x9-inch baking dish with nonstick cooking spray and set aside.
Bring a large pot of salted water to a boil and cook the manicotti shells according to package directions until almost al dente; drain and set aside.
In a large skillet, melt the butter over medium heat. Add the chopped onion, minced garlic, and minced jalapeño peppers and cook until softened, about 4–6 minutes.
Remove the skillet from the heat and stir in the shredded chicken, cream cheese, and sour cream until smooth and well combined. Let the filling cool about 20 minutes, then stir in 1/3 cup of the salsa.
Using a spoon, carefully stuff each cooked manicotti shell with the chicken mixture and arrange the stuffed shells in the prepared baking dish.
In a small bowl, whisk together the remaining 1 cup salsa and the 8-ounce can of enchilada sauce. Pour 1/2 cup of this sauce mixture into the bottom of the baking dish, then place the stuffed shells on top.
If any chicken filling remains, dollop or spread it on top of the shells. Pour the remaining sauce mixture evenly over the stuffed shells.
Toss the Monterey Jack, sharp cheddar, and Parmesan cheeses together and sprinkle evenly over the top of the manicotti and sauce.
Bake in the preheated oven for 35–40 minutes, until the sauce is bubbly and the cheese is browned. Let rest a few minutes before serving.